Effectiveness of Different Teaching Resources for Forming the Concept of Magnitude in Older Preschoolers with Varied Levels of Executive Functions

Background Studies have shown the great importance of early mathematical development as a predictor of subsequent success, which poses the question of how to organize preschool mathematical education with a view to the children’s age characteristics, including their cognitive development. In other words, mathematical concepts and actions should be formed with the help of teaching resources appropriate to the child’s development. Objective To determine the effectiveness of three teaching resources (examples, models, and symbols) in formation of the concept of magnitude in older preschoolers (ages 6–7) with different levels of executive function. Design Four training programs (with 15 twenty-minute lessons each) were developed and conducted in a formative experiment for older preschoolers with different levels of development of executive functions. The lessons addressed the concept of magnitude (length, area, volume), using different types of teaching resources: exemplars (in traditional and game variants), models, and symbols. The total sample of 116 subjects (44% boys) was divided into 4 groups for each of the programs, plus a control group in which no sessions were conducted. The groups were equalized according to the initial level of development of concepts of magnitude and the level of development of executive functions. Results There was a statistically significant increase in the quality of mastery of the concept of magnitude in three experimental groups (“symbolic,” “traditional,” and “traditional with imaginary characters”) compared with the control group. The formative effect of the “model-building” program showed no significant differences from the effect of the child’s natural development (the control group). We also showed that children with a low level of regulation learned mathematical concepts more effectively with the “symbolic” program; children with a medium level of regulation with the “symbolic” and any variant of the “traditional” program; and children with a high level of regulation with the “symbolic” and “model-building” programs. Conclusion The findings underline the importance of both the type of teaching resources used and the level of development of voluntary regulation, when teaching mathematics to preschoolers.


Introduction
Research in psychological and pedagogical science has demonstrated a signi cant in uence of early mathematical skills on subsequent academic and social success, both in school and in adulthood (Jordan et al., 2009;Tikhomirova, 2021;Watts et al., 2018). e great importance of preschool mathematics teaching for subsequent development (Ritchie & Bates, 2013;Watts et al., 2014) indicates the need for organization of e ective education at this age based on the speci cs of child development in the preschool period. is is extremely important given the nature of modern preschool childhood: increasing demands on children's intellectual development, and the discrepancy between children's age requirements and characteristics and some models of education, leisure activities, and children's products (Smirnova, 2019).
Following the cultural-historical and activity approaches, studies have shown (Aleksandrova, 2013;Davydov et al., 1996;Davydov & El'konin, 1966;Obukhova, 1972;Shinelis & Sidneva, 2020) that it is advisable to start introducing children to mathematical reality by mastering an elementary mathematical concept such as magnitude. Magnitude itself is usually determined using three comparison relations (a = b, a > b, a < b); examples of magnitudes that preschoolers constantly confront include length, area, volume, and quantity. e concept of magnitude is essentially a system-forming concept that underlies the concepts of number, function, and gure, and, accordingly, links three domains of mathematics: arithmetic, algebra, and geometry (Davydov, 1962).
According to P.Ia. Galperin's theory of planned stage-by-stage formation of mental actions and concepts, any concepts new to the child should be learned as reference points for relevant actions, revealing the cultural and historical conditions for the origin of these concepts (Gal'perin, 1975). In studies conducted according to the theory of developmental learning and the theory of planned stage-by-stage formation of mental actions and concepts, it has been shown that from the psychological and logical/subject-related standpoints, the most complete and appropriate idea of magnitude is formed when learning actions of comparison (establishing the correspondence or non-correspondence of magnitudes) and measuring quantities using a conditional measure (how many times the measuring instrument ts into the given magnitude) to establish relationships between them (Davydov, 1962;El'konin, 1963;Frolova, 1963;Gal'perin & Georgiev, 1960). e use of a conditional measure makes it possible, rst, to compare objects that cannot be directly placed upon each other, and second, to concretize the relationship between magnitudes and understand how much one magnitude is larger or smaller than another. When teaching is organized this way, the concept of "magnitude" is mastered as a necessary reference-point for a speci c object-oriented action -the action of comparison and measurement (Gal'perin, 1976) -which allows us to say whether children have understood its essential features. However, even when the children perform the same actions, the organization of speci c cognitive situations and tasks can be provided by various teaching methods. is raises the issue of the e ectiveness of such methods.

Teaching Resources and Age Characteristics of Preschoolers
Teaching resources are de ned as anything that facilitates the transfer of knowledge in the instructional process (Salmina, 1988). is may include materials and study aids used in the classroom, the teacher's narrative, etc. ey may di er in form and content. However, the key point in achieving developmental e ects is the di erence among teaching resources according to their function in children's action (Salmina, 1988). From this point of view, it is essential to consider teaching resources that: 1. Establish a meaningful purpose for the child's action; 2. Provide a way for that action to be performed.
In this context, the term "resource" is used here in the sense of an instructional approach. However, the teaching resources used in the functions mentioned above may also be considered as psychological methods that allow one to master new types of activity (Vygotsky, 2004). Mastering the cultural system underlying one's own cognitive activity is an important aspect of the cognitive development of a preschool child (El'konin, 1989;Karabanova, 2005;Venger, 1986). Accordingly, the child's success in mastering mathematical content will depend on the appropriateness of the instructional approaches that are selected. ese instructional approaches and resources must rst of all be consistent with the logic of ampli cation (Zaporozhets, 1986). In other words, the tools should "grow out of " children's natural activities, in which zones of proximal development (ZPD) are also created. Based on the features of children's activities described in various studies (El'konin, 1978;Sarama & Clements, 2009;Shapovalenko, 2004;Shiiaan et al., 2021;Solovieva et al., 2021;Venger & Kholmovskaia, 1978), we have identi ed three possible types of teaching resources for this age group: 1. Exemplars (instructions or rules that are accepted by general agreement of the players), most actively used in games that have rules; 2. Models (diagrams, maps, plans, and other objects that allow the child to display the essential relationships between objects) encountered in children's productive activities (model-building, construction, drawing, etc.); 3. Symbols (a magic wand, an imaginary letter, etc., in which the child singles out and maintains significant relationships through an emotional attitude to the situation being created), which are an essential part of the content of plot role-play.
In order to test the e ectiveness of each type of teaching resource, we developed four di erent approaches to designing a curriculum to teach the concept of magnitude to older preschoolers: 1. Traditional approach with imaginary characters (the key teaching tool here is the exemplars introduced through imaginary characters); 2. Traditional approach (the key teaching tool is exemplars); 3. Model-building approach (the key teaching tool is models); 4. Symbolic approach (the key teaching tool is the symbol); Although the teaching resources identi ed here have analogues in the free activity of most preschoolers, we believe that such tools can play di erent roles, depending on the particular characteristics of the child's development. Voluntary self-regulation plays a key role in this development. us, for example, orientation to exemplars and rules appears signi cantly later than symbolization and model-building (these are the essence of mastering the substitutive function of game objects); such an orientation, closely related to voluntary regulation, appears only at the stage of already rather developed plot role-play (El'konin, 1978;Veraksa & Veraksa, 2016). erefore, it seems to us fundamentally important to consider the e ectiveness of various means of forming mathematical representations in the context of the development of voluntary self-regulation in preschoolers.

Mathematical Development and Voluntary Self-Regulation at Preschool Age
When we speak of self-regulation in this paper, we rely on the concept of regulatory or executive functions (EFs) as developed by A. Miyake and colleagues (Miyake, 2000). According to this concept, EFs are a group of cognitive processes that provide for purposeful problem solving and adaptive behavior in new situations (Diamond, 2012); that is, they are metacognitive capabilities (Morosanova et al., 2021). Executive functions comprise three components: 1) working memory (visual and verbal); 2) cognitive exibility (focusing attention and/or switching attention under changing conditions); 3) inhibitory control (the ability to suppress an impulsive reaction).
Research shows that EFs predict the future performance of preschoolers (Duncan et al., 2007) and are correlated with mathematical ability (Bull & Scerif, 2001;Clements et al., 2016;Jarvis, 2003). us, for example, inhibitory control and cognitive exibility in preschool children are predictors of mathematical ability at an older age (Best et al., 2011;Espy et al., 2004). A low level of EF is associated with di culties in mastering mathematical concepts (Ribner, 2020;Swanson, 2001). ese results raise the question of the particular ways that mathematical education should be organized for children with di erent levels of development of executive functions.

Research Hypotheses
We have suggested that symbolic representations are most appropriate in the preschool period, as these are more natural from the standpoint of preschoolers' play activity, and contribute to self-expression (Veraksa & Veraksa, 2016). Techniques using exemplars are more speci c to traditional schools, where the teacher serves as a conveyor of cultural models and most o en passes them on to students in a directive form. Modeling tools may be di cult to master in that situation. On this basis, we advanced the following empirical hypotheses: 1. Children with a low level of development of EFs will learn mathematical concepts and skills more successfully when using symbolic representations. is hypothesis is based on the fact that symbolization at an early age greatly facilitates perception of the conditions of tasks (Veraksa et al., 2014;Veraksa et al., 2020) and will be the most e ective means of their formation due to the symbolic nature of mathematical representations (Salmina, 1988).
2. Preschoolers with a high level of EF development will show the best results from the model-building approach. is is because visual models are more conducive to the cognitive development of children with a pronounced cognitive orientation (Venger, 1995).

Participants
e sample comprised 150 children aged 6-7 (mean age 6.9) attending kindergarten preparatory groups, of whom 65 were boys (43%) and 85 were girls (57%). e study was conducted in the 2021-2022 academic year. All participants attended a Moscow educational complex where the program "From Birth to School" was used as the base program. e study was approved by the Ethics Committee of the Faculty of Psychology of Lomonosov Moscow State University.

Questionnaires
Methods for Mastering the Concept of Magnitude A diagnostic toolkit was developed for use with preschool children to assess the quality and stability of their formation of elementary mathematical representations of magnitudes and their relationships.
is diagnostic technique included four types of tasks for each magnitude: length, area, volume. e children were given two tasks to solve for each magnitude, about the ability to compare objects; they were asked to select an object the same size as another one -for example, to nd the rectangle with the same length as that shown in a drawing. Two tasks each for the ability to use a measuring instrument correctly ("who measured correctly?"): to apply it so that there is no empty space between measurements, to use equal measuring instruments, etc. Two tasks each for actually measuring a magnitude with a conditional measuring instrument and recording the result with labels or a number ("how many times does the measuring instrument t in this magnitude?"). Two tasks each to understand how the number depends on the measuring instrument used (the larger the measuring instrument, the fewer times it ts into the magnitude). Two more assignments were included for making sets ("what would be le over if such and such sets were used?"). e ability to put together complete sets was not speci cally targeted in this study during the lessons, so we considered these tasks to be within the children's zone of proximal development (ZPD). e tasks for making sets were assessed depending on the amount of assistance provided to the child by the tester and the correctness of the answer: the preschooler received 2 points for correctly solving the task independently, 1 point for solving the task correctly with the help of the tester's prompts; 0 points if the task is not solved or is solved incorrectly even with the help of an adult. e formation of concepts of length and area was assessed from 0 to 10 points; of volume from 0 to 7 points; and for tasks in the ZPD from 0 to 4. e total possible score was 31 points. Diagnosis of mathematical concepts and skills was performed individually with each child.

Methods for Assessing the Development of Executive Functions
Recent studies show that the level of executive functions is signi cantly associated with children's success in mastering mathematical content (Clements et al., 2016;Veraksa et al., 2020); therefore, we used the level of development of executive functions as the criterion for dividing the children into groups. To measure students' EF, we used the NEPSY-II subtests (Korkman et al., 2007) for visual memory (Memory for Designs) and auditory working memory (Sentence Repetition), inhibition and switching (Naming and Inhibition, Statue), and also cognitive exibility ( e Dimensional Change Card Sort) and visual-spatial memory (Schematization). is allowed us to measure various components of preschoolers' cognitive processes. e diagnosis of EFs was performed with children on an individual basis during two meetings with each child. e NEPSY-II Memory for Designs subtest was used to assess working visual memory. is test includes the following nal scores: content scores are awarded for correctly remembering picture details (maximum 46 points); spatial scores re ect how correctly the child remembers the con guration of a picture (maximum 24 points); and bonus scores are awarded to the child for correctly remembering and looking at both dimensions simultaneously (maximum 46 points). e three indicators are summed up in the nal score (maximum 116 points).
Verbal working memory was assessed using the NEPSY-II "Sentence Repetition" subtest, which consists of 17 sentences that gradually become more di cult to remember due to their length and grammatical structure. Children receive 2 points for each sentence they repeat correctly; one point if they make one or two mistakes in the repetition by skipping, replacing or adding words, or changing the order of words; and if the child makes three or more mistakes or does not answer, no points are awarded. e assignment is terminated if the child receives 0 points four times in a row. e NEPSY-II "Naming and Inhibition" subtest assesses information-processing speed and inhibition of impulsive reactions. It consists of two blocks: a series of white and black circles and squares and a series of white and black arrows showing di erent directions (up and down). Two tasks were performed with each series of pictures: a task to identify the form (in this case, the child simply had to quickly name the forms that he saw) and an inhibition task. In the latter case, the child had to do everything contrariwise: for example, if he saw a square, he was supposed to say "circle" and so on. For each task, the researchers recorded the number of mistakes the child made and corrected or could not correct, as well as the time it took to complete the task.
" e Dimensional Change Card Sort" test (Zelazo, 2006) was used to assess cognitive exibility. is technique consists of three tasks for sorting cards. First, the children must arrange the cards by color, shape, and then follow a complex rule: if the card has a frame, they must sort it by color, and if there is no frame, they must sort it by shape. For each correctly sorted card, the child receives 1 point; at the end, the number of points for each series is calculated (maximum 6, 6, and 12 points, respectively), and then the total score for all tasks is calculated (maximum 24 points).
We used the "Statue" subtest (NEPSY-II, Korkman et al., 2007) to assess "hot" self-regulation and physical inhibitory control. In this test, the child is instructed to remain motionless with eyes closed for 75 seconds, inhibiting impulsive reactions to distracting sounds. An assessment is performed for each 5-second interval: 2 points are awarded if the child did not make any mistakes in the 5-second interval, 1 point is given if 1 mistake was made, and 0 points if 2 or more mistakes were made. Large movements of the arms, body, legs, head, opening of the eyes, vocalization or laughter are all considered errors. e total score (max. 30) and the number of errors are calculated for three categories: movements, eyes, and sounds.
We used the "Schematization" technique to assess planning and checking and visual-spatial orientation. Here, children are asked to nd a "route" through an extensive system of streets, using the notation of this route with the help of a diagram and/ or a conditioned image in the form of a system of landmarks. e child has to take into account the sequence of landmarks and/or the direction of turns.

Methods for Assessing Intellectual Development
As a supplement, non-verbal intelligence was diagnosed using Color Progressive Matrices by D. Raven (Raven & Kort, 1997). In this technique, the children need to choose one of the six proposed images in order to complete a drawing while following a certain logic. e technique contains three series of 12 tasks (maximum score -36).
All the techniques were presented to the children in digitized form with a mobile app.

Procedure
e study comprised several stages. First, the cognitive processes of children were assessed using the NEPSY-II subtests "Card Sorting" and "Schematization, " as well as D. Raven's matrices. A er evaluating their EF, the children were divided into three subgroups by level of cognitive development (low, medium, high) according to the results of cluster analysis (K-means clustering) performed in IBM SPSS Statistics 22. Before the beginning of the experimental sessions, a pre-test of mathematical concepts and skills was conducted using the authors' diagnostic tools.
Next, participants from each subgroup with low, medium, and high EF levels were randomly assigned to four experimental groups and one control group, so that the ratio of participants in the groups was uniform. For each approach, 15 experimental sessions lasting 15-20 minutes were held in mini-groups of 3-4 children. e sessions occurred twice a week in the rst half of the day in the groups at the kindergarten. e control group did not attend any special sessions. e sessions were completed for all approaches simultaneously, a er which a post-test of mathematical conceptions and skills was performed, similar to the baseline diagnostics in the experimental and control groups. A month a er the experimental sessions, some of the children took part in a delayed post-test.

Formative Sessions
e programs we developed for the formation of concepts of magnitudes corresponded to the types of methods that "grow out of " the natural activity of preschool children.
In the rst program ("traditional"), an exemplar was o ered as the main instruction for performing actions. e children were given speci c instructions on how to perform an action as a way of solving a problem (for example, "here is how you measure with a ruler, " "look how I do it, " etc.). e meaningfulness of the tasks was not speci cally addressed; the children were presented with tasks such as "measure, " "compare, " " nd one that's the same. " Why this had to be done was not discussed. Note that analysis of modern Russian programs and mobile apps for preschoolers has shown that the main teaching resource used is the exemplar, a rule taught to the children for performing a task or action by demonstration of the action, while the need to use the mathematical concepts and actions is given as an external condition Sidneva et al., 2021).
In the second program ("traditional with imaginary characters"), the concept of magnitude was introduced in exactly the same way as in the rst program: through a directive instruction about the mode of action and tasks that did not disclose the meaning of this action. However, the tasks given to the children were presented by characters in a game (Dumbo, Wizard, etc.), depicted in a colorful picture. ese characters did not perform a symbolic function, nor did they help to make the task more meaningful. ey were an external game element, introduced in order to evaluate the role of this type of game element, while maintaining the basic orientation to the exemplar.
In the third program ("model-building"), we introduced the concept of magnitude and worked on it through design tasks ("choose a suitable column for the building, " "what kind of tiles can be used to lay the oor in the bathroom?"). Here, meaningfulness was determined by, on the one hand, a real everyday or engineering situation that needs to be resolved, and on the other, the child's desire to act like an adult, for example, like an engineer or like Dad, who repairs things. e solution was introduced as something that could help solve this type of problem. When actions with real objects were di cult (for example, they are too heavy, big, or fragile, or you need to perform the action right the rst time so as not to have to redo the repair, etc.), the problem could be solved by constructing a model of the real objects (a diagram, drawing, or other type of model) and using it to test hypotheses. Using various models and schematized methods, the children were able to learn generalized information about the essential properties of the real world (Shapovalenko, 2004). And research has shown that it is indeed through manipulation of such models, that both the development of initial mathematical concepts and the formation of the main mental neoformations take place (Venger, 1978).
In the fourth program ("symbolic"), the concept of magnitude was introduced and worked out through tasks of helping an imaginary character (for example, "pour the same amount of the water of life to save the queen, " "help Winnie the Pooh nd his way home from the dark forest") that created emotional meaningfulness for the goal of the action. e means of solving the problem situation were also symbolic objects (for example, a magic ball for measuring a route; umbrellas that can protect a drawing on asphalt from rain; a magic cup). ese symbolic representations established the problem situation, key points of orientation, and relationships for its solution, becoming reference points for mastering the concept of magnitude. In this case, the children did not need a model of the action; they themselves could construct the necessary action based on the symbolic image of the situation, since the symbol as a cognitive tool facilitates perception of the conditions of the task in a situation of uncertainty (Veraksa et al., 2014;Veraksa et al., 2020 ). And at the same time, it ensures the children's emotional involvement in the activity (Leont' ev, 2000;Veraksa et al., 2015) e programs we developed are identical in terms of the object-speci c actions performed by the children: measurement and selection with the help of conditional measures, but they di ered in the teaching resources used. e following concepts were chosen as formed concepts in all the programs: length (including width and height), area, and volume.
Experimental sessions were conducted by specially trained teachers, who did not themselves perform the preliminary and subsequent testing.

Results
e nal sample of the formative experiment included 116 preschoolers who had gone through both EF diagnostics and a pre-test in mathematics. e children who were included in the formative experiment did not di er from those excluded from it in the development of their visual working memory (Chi-square = 0.9, p = 0.3), inhibition and switching (Chi-square = 0.3, p = 0.6), cognitive exibility (Chi-square = 1.9, p = 0.16), visuospatial memory (Chi-square = 0.01, p = 0.9), and verbal intelligence (Chi-square = 0.2, p = 0.6). However, di erences were found in their auditory-verbal working memory (Chi-square = 5.7, p = 0.01), which we did not consider signi cant, since research has shown that auditory-verbal working memory is weakly associated with mathematical development (Bull & Johnston, 1997) e participants comprised 51 boys (44%) and 65 girls (56%). No signi cant statistical di erences were found in the level of mathematical abilities for boys and girls (Mann-Whitney U test, p > 0.05).
Before further analysis, we will provide statistics on the distribution of children by approach in the formative experiment.
Descriptive Statistics e preschoolers' EF level of development was taken into account when they were distributed according to the experimental conditions (see Table 1). Four experimental groups ("Traditional, " "Traditional with imaginary characters, " "Model-building, " "Symbolic") and one control group were equalized in terms of the level of the children's executive functions (Pearson's Chi-square, p > 0.05). No statistically signi cant di erences were found in the distribution of children based on EF level by approach. Children with di erent EF levels were divided proportionally into the experimental groups. In that way, further analysis of group di erences is justi ed without taking into account the limitations in this part. Also, none of the identi ed groups di ered in the pre-testing for mathematics (see Table 2). e lack of di erences in the pre-testing is a prerequisite for correct interpretation of the results of the formative in uence. A er the formative part of the experiment, only children who had attended more than half of the sessions were included in the subsequent study. Eighty 80 children were tested in the post-test, and 44 in the delayed post-test. Although the number of children who participated on the post-test was lower than on the pre-test, the proportional distribution of EF levels within the experimental conditions was maintained (see Table 3). e groups remained equal according to this criterion, which removes further limitations on data analysis. ere were 44 children in the delayed post-test: 12 children from the control group, 9 from the "Symbolic" approach, 11 from the "Model-building" approach, and 6 from the "Traditional" approaches.

Analysis of the E ectiveness of Teaching Resources in the Formation of Concepts of Magnitudes
In order to assess the e ectiveness of the formative sessions, we performed a nonparametric statistical analysis of the nal scores of the pre-and post-tests for the experimental and control groups. e comparison showed signi cant di erences in the total score for diagnostics of mathematical ability between the results of the pretest and post-test, and between the pre-test and delayed post-test (Wilcoxon Z test, p < 0.05). ere were no signi cant di erences found between the post-test and delayed post-test (Wilcoxon Z test, p > 0.05), which may suggest some stability in the results of the formative sessions (Wilcoxon Z test, p < 0.05). e minimum overall post-test score was in the control group -2 points out of 31, with a minimum score of 7.5 for the experimental conditions. us, the results of both post-tests were signi cantly higher than the pre-test results.
We also note that the maximum score for diagnostics was indeed achieved on the post-test, with 29 points, and the maximum score for the delayed post-test, 27.5, demonstrates a slight decline. Since no signi cant di erences were found between the post-test and delayed post-test, and the post-test had the larger variation in total score (2 to 29 for the post-test; 4.5 to 27.5 for the delayed post-test) and a larger sample, the post-test results will be considered for further analysis. Table 4 Descriptive statistics of the nal test score for the experimental and control groups.

M SD Min Max
Experimental groups To assess the e ectiveness of speci c teaching resources, we compared the increase in the nal score on the post-test for each approach (see Table 5, Figure 1). We found that the type of formative lesson does indeed have a signi cant impact on the increase in the overall score for diagnosis of mathematical concepts and skills (ANOVA with non-parametric Welch correction, p < 0.05 with equality of variances, Levene's criterion, p > 0.05). Pairwise comparison of the increase in mean values for di erent approaches to formation showed that children who studied according to the "Symbolic" and "Traditional with imaginary characters" programs showed a signi cantly greater increase in total score on the post-test compared with the control group (LSD, p < 0.05). Signicant di erences were also found between children from the control group and chil-dren in the "Traditional" program (Mann-Whitney U-test, p < 0.05). e formative e ect of the "Model-building" program does not show signi cant di erences from the natural development of a child attending kindergarten (the control group) (see Table 5). We also note that children who attended formative sessions with the "Symbolic" approach showed a signi cantly greater increase in scores than those from the "Model-building" approach (LSD, p < 0.05). Table 5 Analysis of multiple comparisons of approaches by increase in post-test score  For individual mathematical representations and actions, the following results were found: 1. The participants in the "Symbolic" approach showed a significantly greater increase in post-test scores than the children from the control group, in terms of the formation of concepts of length and area, the dependence of the number on the measurement; and the ability to select values and assemble "complex" sets (LSD, p < 0.05); 2. Children following the "Symbolic" approach also differed significantly in their formation of the concept of the dependence of the number on the measurement, from the children in the "Model-building" and "Traditional with imaginative characters" approaches (LSD, p < 0.05); 3. The increase in the total score for area and the ability to select magnitudes, in children who attended formative sessions according to the "Traditional with imaginative characters" program, was significantly greater than in children from the control group (LSD, p < 0.05); 4. There were no significant differences in the increase in the score on the posttest for volume, or the ability to use a measuring instrument to measure magnitudes (LSD, p > 0.05).

Analysis of the E ectiveness of Teaching Resources Depending on the Level of EF
An analysis of variance with repeated measurements was performed to test the hypothesis that children with a lower level of EF will most e ectively master the mathematical concepts of magnitudes when they are taught with the "Symbolic" approach. Table 6 shows the di erential characteristics of di erences between baseline test scores (pre-test) and subsequent ones (post-test). e joint interaction of the two factors -approach and level of EF -has a statistically signi cant e ect on the increase in the total score on the post-test (ANOVA with nonparametric correction, p < 0.05). Looking at the assessment of the mean values of di erential di erences, it is important to note that the greatest increase in scores among students with low EF was found in the formation of concepts of magnitude in the symbolic approach. e mean increase in the score of children with low EF in the symbolic approach was more than 11 points, with an overall average increase of 3.94, and this was the high-est compared to other groups. We emphasize that children with low EF who did not attend formative sessions showed a tendency toward some decrease in scores on the post-test compared to the pre-test and showed the smallest increase compared to the other groups.
To demonstrate the di erences more clearly, Figure 2 shows pro le plots for estimating the mean values of the di erential di erences. However, children with a low level of self-regulation (executive function) mastered mathematical representations when following the "symbolic" approach signi cantly better than children from the control group and the "traditional with imaginary characters" approach (Welch's t-test, p = 0.001 and p = 0.01, respectively). Preschoolers with a medium level of EFs mastered the material better when taught using both traditional programs than did the children from the control group and those taught according to the model-building approach (Welch's t-test, p < 0.05 for each pair), and children with a medium level of self-regulation who studied with the "symbolic" program showed a signi cantly greater increase in scores than the control group (Welch's t-test, p = 0.015). Participants with a high level of EF mastered the mathematical content in the Model-building approach more successfully than did the children from the Traditional with Imaginary Characters approach (Welch's t-test, p = 0.004).
at said, we emphasize that, in general, the children with a high level of regulation showed a signi cantly greater improvement in mastery of mathematical concepts than those with a medium or low level (LSD, p < 0.05). Even in the control group, the scores of these children increased by an average of 6 points. On the other hand, children with low and medium levels of EF from the control group showed the least increase and even a worsening of results on the post-test. Pupils with a medium level of development of self-regulation who were taught with the "model-building" approach also showed an increase by an average of 1 point; that is, the level of formation of their concepts of magnitudes did not actually change.

Discussion
An important result of this study was the signi cant increase in scores on the posttest compared to the pre-test, which indicates the e ectiveness of our sessions. We have also seen that this e ect has some stability, since a month a er the experiment, the children completed the test just as successfully as they had immediately a er the completion of the formative sessions. Besides the general developmental e ect of the formative sessions, we also obtained signi cant di erences among the approaches, and, accordingly, among the teaching resources used in them. e use of symbols and exemplars to form concepts of magnitude turned out to be the most productive and successful, whereas learning based on models did not di er in its impact from children's natural development and what is learned from standard kindergarten classes. We attribute this result to the fact that operating with abstract signs of objects requires children to have quite developed visual-e ective thinking and an internal plan of action (Venger, 1995). erefore, for most children, this program may lie outside the zone of their actual and proximal development, since research has shown that the formation of abstract thinking only begins at the end of the preschool years (El'konin, 1989;Karabanova, 2005). We also emphasize that only the children who had been taught according to the "symbolic" approach coped with the making sets task (this action was not specially formed) signi cantly better than the control group, which once again indicates the e ectiveness of the symbol as a teaching tool in expanding the child's ZPD. us, our rst hypothesis about the di erent e ectiveness of the approaches was partially con rmed.
It is interesting that we did not nd any increase in scores that di ered from natural development in our testing for the concept of volume, the ability to use a measuring instrument, and to measure magnitudes. at is, formative sessions did not help children to master these actions speci cally for a magnitude such as volume. Why is that? Volume is the most complex of the formed magnitudes. It is known that understanding of the conservation of volume arises much later than the conservation of length and quantity (Piaget, 1994). e formation of this concept is more di cult and occurs more slowly in preschoolers than the concepts of length and area (Obu khova, 1972), and some studies have shown that this concept does not lie within the ZPD of a preschooler . For volume as a magnitude, the strongest visual attributes (for example, the height of water in a jar, the width of a ask, etc.), prevent one from "grasping" the essential characteristics and fully mastering the concept. However, this result may be associated with incorrectly selected guidelines and actions for this concept in the programs and with methodological inaccuracies.
From analyzing the results, it can be asserted that the initial level of development of cognitive processes in older preschoolers is directly related to the children's ability to learn new mathematical concepts. Children with a high level of EF are more successful in mastering new mathematical skills and concepts than those with a low and medium level of development of executive functions. is is consistent with earlier research ndings that the level of EF is a predictor of the development of mathematical skills (Bull & Scerif, 2001;Clements et al., 2016;Jarvis, 2003;Veraksa et al., 2020). e fact that children with a low level of EF scored the highest increase in points a er completing their instruction according to the "symbolic" approach, and children with a high level of EF according to the "modeling" approach, may indicate the importance of choosing suitable tools for forming mathematical concepts, considering the level of formation of cognitive processes for student development (Clements et al., 2016;Veraksa et al., 2020) and con rms our second and third hypotheses.
It should also be emphasized that preschoolers with low EF who did not attend any special sessions showed worse results on the second test, which suggests the importance of special work with this group of children.

Conclusion
We see the possibility of e ective teaching even of children with a low level of voluntary self-regulation, by using a symbolic approach, whereby new ideas and concepts are introduced in a special emotionally constructed form, which makes it possible to motivationally include the children in learning and simplify their perception of the task (Veraksa et al., 2014). is approach is based on the use of symbols as a special form of mental representation of an object. Children use symbols in play as a means of self-expression (Veraksa et al, 2016). Mathematical instruction can also be symbolized to increase the e ectiveness of the learning of children with low EF, whereas for children with high levels of cognitive development, more actions can be included in their mathematical instruction to operate with visual models, schematized representations of objects to stimulate their cognitive development (Venger, 1995).

Limitations
Finally, we note that the limitations of this study include the experimenter e ect, which could a ect the results of the formative sessions and the entire study, as well as the relatively small number of children in each of the subgroups at the time of the post-test.

Ethics Statement
e study was approved by the Ethics Committee of the Faculty of Psychology of Lomonosov Moscow State University.

Informed Consent from the Participants' Legal Guardians
Participants gave informed consent before taking part.

Author Contributions
A.N. Veraksa and A.N. Sidneva conceived of the idea and supervised the project, A.N. Sidneva and V.A. Plotnkova developed the design of the experiment. M.S. Aslanova made a statistical analysis. All authors discussed the results and contributed to the nal manuscript.

Con ict of Interest
e authors declare no con ict of interest.